PHYSICAL REVIEW A 82, 042306 (2010)

Performance comparison of dynamical decoupling sequences for a qubit in a rapidlyfluctuating spin bath

Gonzalo A. Alvarez,1,* Ashok Ajoy,1,2,3 Xinhua Peng,1,4 and Dieter Suter1,†1Fakultat Physik, Technische Universitat Dortmund, D-44221 Dortmund, Germany2Birla Institute of Technology and Science, Pilani, Zuarinagar, Goa 403726, India

3NMR Research Centre, Indian Institute of Science, Bangalore 560012, India4Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,

University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China(Received 11 August 2010; published 7 October 2010)

Avoiding the loss of coherence of quantum mechanical states is an important prerequisite for quantuminformation processing. Dynamical decoupling (DD) is one of the most effective experimental methods formaintaining coherence, especially when one can access only the qubit system and not its environment (bath). Itinvolves the application of pulses to the system whose net effect is a reversal of the system-environment interaction.In any real system, however, the environment is not static, and therefore the reversal of the system-environmentinteraction becomes imperfect if the spacing between refocusing pulses becomes comparable to or longer than thecorrelation time of the environment. The efficiency of the refocusing improves therefore if the spacing betweenthe pulses is reduced. Here, we quantify the efficiency of different DD sequences in preserving different quantumstates. We use 13C nuclear spins as qubits and an environment of 1H nuclear spins as the environment, whichcouples to the qubit via magnetic dipole-dipole couplings. Strong dipole-dipole couplings between the protonspins result in a rapidly fluctuating environment with a correlation time of the order of 100 µs. Our experimentalresults show that short delays between the pulses yield better performance if they are compared with the bathcorrelation time. However, as the pulse spacing becomes shorter than the bath correlation time, an optimum isreached. For even shorter delays, the pulse imperfections dominate over the decoherence losses and cause thequantum state to decay.

DOI: 10.1103/PhysRevA.82.042306 PACS number(s): 03.67.Pp, 03.65.Yz, 76.60.Lz

I. INTRODUCTION

Quantum mechanical systems have an enormous potentialfor realizing information processing devices that arequalitatively more powerful than systems based on classicalphysics [1]. The main requirement for quantum informationprocessing (QIP) is that the system evolves according to theSchrodinger equation, under the influence of a Hamiltonianthat is under precise experimental control. However, no systemis completely isolated, and disturbances from its surroundingenvironment (bath) spoil the quantum identity of the system.This process is often called decoherence [2] and limits the timescale over which quantum information can be retained andthe distance over which it can be transmitted [3–5]. Reducingthe effects of decoherence is therefore one of the mainrequirements for reliable quantum information processing.Several protocols have been developed for quantum errorcorrection [6,7]; however, they prove advantageous only forlow levels of environmental noise.

In all existing experimental architectures for QIP the noisebackground is too large, and this limits the applicabilityof these protocols. A promising technique for reducingthe noise to a level where error-correcting codes can takeover is called dynamical decoupling (DD) [8,9]. It aimsto reduce the interaction of the system with the environ-ment through control operations acting only on the system.It requires relatively modest resources, since it requires

*galvarez@e3.physik.uni-dortmund.de†Dieter.Suter@tu-dortmund.de

no overhead of information encoding, measurements, orfeedback.

Although the mathematical framework of dynamical de-coupling was introduced fairly recently [8], the 6-decade-oldHahn NMR spin-echo experiment [10] can be considered theearliest and simplest implementation of this method. It consistsof the application of a π pulse to a spin qubit ensemble at timeτ after the spins were left to undergo Larmor precession ina magnetic field. This effectively reverses a pure dephasingsystem-environment (SE) interaction, i.e., one that does notcause a net exchange of energy between the system andthe bath. The combined effect of the evolution before therefocusing pulse and a second period of the same duration afterthe pulse vanishes. Physically, the dephasing and rephasing ofthe spins can be observed as an apparent decay of the averagemagnetization in the system and a subsequent increase afterthe refocusing pulse (a spin echo).

The Hahn echo can (for an ideal pulse) completely eliminatethe interaction with the environment, provided it is timeinvariant. In practice, this is often not the case, and a changein the environment reduces the refocusing efficiency [10,11].To reduce the problems due to a time-dependent environment,Carr and Purcell suggested replacing the single pulse of theHahn echo by a sequence of pulses at shorter intervals (the CPsequence) [11], thus reducing the changes in the environmentbetween successive pulses. For sufficiently short pulse inter-vals, elimination of system-environment interactions becamepossible even in a time-dependent environment. However,the increased number of pulses led to another problem: Ifthe refocusing pulses are not perfect, they actually becomea source of decoherence (and thus signal loss) instead of

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ALVAREZ, AJOY, PENG, AND SUTER PHYSICAL REVIEW A 82, 042306 (2010)

eliminating it. This problem was significantly reduced bya simple modification of the CP sequence: if the rotationaxis of the refocusing pulses is parallel to the initial spinorientation, the effect of pulse errors is significantly reducedover a cycle [12]. This is known in literature as the Carr-Purcell-Meiboom-Gill (CPMG) sequence.

In the context of QIP, there has been renewed effort ineliminating the effects of the system-environment interactionthat lead to the loss of quantum information. For this, it is oftenimportant that the refocusing reduces the effect of the system-environment interactions by several orders of magnitude. Inaddition, the effect of pulse errors must be minimized, andthe sequence has to work for all possible initial states ofthe system. Several pulse sequences that achieve this wereintroduced [8,13–15], which consist of periodic sequences ofpulses; they were thus called periodic DD (PDD). By design,they allow one to decouple the system from the environmentfor a general SE interaction, i.e., one that causes dephasing aswell as dissipation.

Experimentally, DD is achieved by iteratively applyingto the system a series of stroboscopic control pulses incycles of period τc. Over that period, the time-averaged SEinteraction Hamiltonian vanishes. The time average over τc

can be calculated using average Hamiltonian theory [16]. Ifthe average Hamiltonians are calculated by a series expansion,such as the Magnus expansion, improving the pulse sequenceusually corresponds to progressively eliminating higher-orderterms in the expansion. Khodjasteh and Lidar [17] introducedconcatenated DD (CDD) as a scheme that recursively generateshigher-order DD sequences for this purpose. Here, the lowestlevel of concatenation is a PDD sequence. The improvementachieved by concatenation comes at the expense of anexponential growth [17] in the number of applied controlpulses. In contrast, for the case of a pure dephasing or puredissipative interaction Hamiltonian [18,19], Uhrig developed asequence (UDD) [20] that reduces higher orders in the Magnusexpansion with only a linear overhead in the number of pulses.Unlike other DD sequences, in the UDD sequence the delaybetween successive pulses is not equal, i.e., the pulses are notequidistant. In the limit of a two-pulse cycle, UDD reduces tothe CPMG sequence. Recent proposals of DD sequences thatare a hybrid between UDD and CDD are predicted to improveDD performance of previous methods [21,22].

The UDD sequence was tested on ion traps [23,24], electronparamagnetic resonance [25], and liquid-state NMR [26] andfound to outperform equidistant pulse sequences, in particularCPMG, for environments with a high-frequency or strongcutoff. CDD sequences were recently tested in solid-stateNMR [27]. However, while some sequences for particularenvironmental noises were tested, a comparison betweensequences for different kinds of environments is still missing.Most of the sequences were designed assuming ideal pulsesand some of them predict to compensate pulse imperfection.However, an experimental test of this aspect is still needed.In parallel to this work recent DD implementations on aqubit interacting with a slowly fluctuating spin-bath weretested [28–32].

Other questions relate to the optimal cycle time: It istheoretically predicted and experimentally demonstrated thatsequences that reduce higher-order terms of the Magnus

expansion perform better than low-order sequences for slowmotion environments with high-frequency or strong cutoff,when the bath correlation time τB is longer than the sequencetime τc (cycle time). However, the strength and duration ofcontrol pulses are limited by hardware, yielding a minimumfor the achievable DD cycle time. As some examples on this di-rection, Viola and Knill [15] and Gordon et al. [33] proposed ageneral method for DD with bounded controls. Khodjasteh andLidar, keeping the delay between pulses constant, predicted anoptimal CDD order for reducing decoherence [34,35]. Biercuket al. [23,24] needed to consider the finite length of pulsesin their simulations, assuming them perfect but producing aspin-lock during their application times, in order to fit themto the experiments. Additionally Hodgson et al. [36], whileassuming instantaneous perfect pulses, theoretically analyzedDD performance constraining also the delay between pulses.They set a lower limit to the delays, making them larger than thepulse duration to satisfy the instantaneous pulse approximationin their theoretical model for experimental conditions. Whenthe delays are strongly constrained, they predict that DDprotocols like CDD or UDD, which are designed to improvethe performance of lower DD orders if the regime of arbitrarilysmall pulse separations is achievable, in general lose theiradvantages. However, experiments are missing in order todemonstrate these predictions and very little is known aboutthe performance of DD sequences under conditions wherethe cycle times are comparable to or longer than the bathcorrelation times. Recently Pryadko and Quiroz approachedthis regime but only for the extreme case of a Markovianenvironment [37].

While the finite length of pulses limits the minimum cycletime reducing the maximal achievable DD performance, theirimperfections also contribute to reducing it. It is well knowthat CPMG-like sequences are too sensitive to the initial statewhen pulse errors are considered [13,14]. A comparisonsbetween the CPMG and UDD sensitivity against pulse errorswas performed in Ref. [24]. Overall UDD was shown to bemore robust against flip angle errors and static offset errors,with the exception that CPMG is more robust for initial stateslongitudinal to the control pulses. But both of them are tooasymmetric against initial state directions. In general, whilesome DD sequences were developed to compensate flip-angleerrors and to have a performance more symmetric againstinitial conditions, an extensive study of their performancefrom a QIP perspective is still missing and additionally isnot done for CDD sequences. For example, an optimal cycletime when considering imperfect finite pulses was predictedby Khodjasteh and Lidar [34].

In this article, we compare experimentally the performanceof different DD sequences on a spin-based solid-state systemwhere the cycle time τc is comparable to or longer than thecorrelation time τB of the environment. Here, the spin(qubit)-system interacts with a spin-bath where the spectral density ofthe bath is given by a normal (Gaussian) distribution. Thesekinds of systems, typical in NMR [38], are encountered in awide range of solid-state systems, as, for example, electronspins in diamonds [30–32], electron spins in quantum dots[28,29,39], and donors in silicon [40,41], which appear tobe promising candidates for future QIP implementations. Inparticular we consider the case where the interaction with the

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bath is weak compared with the intrabath interaction. For oneside the latter point complement and distinguish our work fromthe recent submitted articles [28–32]. For the other the aim ofour work is a comprehensively and detailed comparison of theperformance of different sequences considering different ini-tial states. We find how the performance of the DD sequencesdepends on the initial state of the qubit ensemble with respectto the rotation axis of control pulses with finite precision. Whenthey are in the same direction, the CPMG sequence is the bestDD sequence for reducing decoherence, i.e., it maintains thestate of the ensemble for the longest time. However, if theinitial state of the ensemble is not known, we find that CDDprovides the best overall performance. Stated equivalently,the CDD scheme provides the best overall minimizationof the environmentally driven quantum mechanical evolutionof the system. Additionally, we experimentally demonstrateand quantify the predicted optimal delay times for maximizingthe performance of the respective DD sequences. This impliesthat pulse errors are a limiting factor that must be reduced toimprove DD performances. In general our results complementsome of the previous findings and predictions for some of theexperimentally tested DD sequences and provide new resultsfor untested ones. One of the main message is that a faircomparison of the performance of DD sequences should use aconstant average number of pulses per unit time.

This article is organized as follows. Section II describesthe qubit and bath system used in our experiment, and themechanisms of coupling between them. In Sec. III we give abrief summary of dynamical decoupling and a description ofthe tested sequences: the Hahn echo, CPMG, PDD, CDD, andUDD. Our limited choice of sequences includes those mostaccepted by the QIP community and allows us to discuss themost important points. Section IV contains the experimentalresults and their analysis. In Sec. V we compare the variousDD sequences under the same conditions. In the last sectionwe draw some conclusions.

II. THE SYSTEM

Our system consists of a spin 1/2 (qubit) in a strongmagnetic field oriented along the z axis, interacting with abath consisting of a different type of spins 1/2. The totalHamiltonian in the laboratory frame is

HL = HLS + HL

SE + HLE, (1)

where HLS is the system Hamiltonian, HL

E is the environmentHamiltonian, and HL

SE is the system-environment interactionHamiltonian:

HLS = ωSSz, (2)

HLSE = Sz

∑j

bSj Ijz , (3)

HLE = ωI

∑j

I jz +

∑i<j

dij

[2I i

z Ijz − (

I ix I

jx + I i

y Ijy

)], (4)

where S is the spin operator of the system qubit, the spinoperators I

jx ,I

jy , and I

jz act on the j th bath spin, ωS and

ωI are the Zeeman frequencies of the system spin and thebath spins, respectively, and bSj and dij are the coupling

constants, and we use frequency units (h = 1). In solids, thespin-spin interaction is dominated by the dipolar interaction[38]. Since S and I are different types of nuclei, it is possibleto neglect the terms of the dipolar coupling Hamiltonian thatdo not commute with the strong Zeeman interaction because|bSj |/|ωS − ωI | � 10−4 [38]. The remaining terms have theIsing form (3). Similarly, the homonuclear interaction betweenthe bath spins is truncated to those terms that commute withthe total Zeeman coupling, which we assume to be identicalfor all bath spins.

In the high-temperature thermal equilibrium [38], thedensity operator of the system spin is

ρS,eq. ∝ Sz, (5)

where we consider only the system (S spin) part of the totalHilbert space. We also neglect the part proportional to the unitoperator, which does not evolve in time and does not contributeto the observable signal.

To generate the initial state for our DD measurements, werotate the thermal state to the xy plane by applying a π/2pulse. The resulting state is

ρS(0) ∝ S{ x

y}. (6)

For an isolated spin system this magnetization precessesindefinitely around the static magnetic field at the Zeemanfrequency ωS .

Taking the system-environment interaction into account,the effect of the coupling operator HL

SE is the generationof product terms of the form S±I

jz in the density operator,

correlating the system with the environment. Since we onlyobserve the system part of the total Hilbert space, weeffectively project the correlated system onto this subspace,

ρS = TrI {ρtot}, (7)

where TrI represents the partial trace over the environmentaldegrees of freedom and ρtot represents the density operatorof system plus environment. The result of this projectioncorresponds to a loss of coherence by dephasing. This freeevolution of the system under the SE interaction is called thefree induction decay (FID) in NMR terminology. The decayprocess of the system state is usually called relaxation in NMRterminology or decoherence in quantum information.

In the following, we will describe the dynamics of thesystem in a rotating frame of reference [38]: The systemrotates at the (angular) frequency ωS around the z axis and theenvironment at ωI . As a result, the rotating frame Hamiltonianbecomes

Hf = HS + HSE + HE, (8)

where

HS = HLS − ωSSz = 0, (9)

HSE = Sz

∑j

bSj Ijz , (10)

HE = HLE − ωI

∑j

I jz =

∑i<j

dij

[2I i

z Ijz − (

I ix I

jx + I i

y Ijy

)].

(11)

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This transformation is exact, since the Zeeman terms commutewith all other terms in the Hamiltonian as well as with theequilibrium density operator.

The effect of the environment-Hamiltonian HE on theevolution of the system may be discussed in an interactionrepresentation with respect to the evolution of the iso-lated environment: the system-environment Hamiltonian thenbecomes

H(E)SE (t) = e−iHEtHSEeiHEt

= Sze−iHEt

⎛⎝∑

j

bSj Ijz

⎞⎠ eiHEt . (12)

Since HE does not commute with HSE, the effectivesystem-environment interaction H(E)

SE becomes time depen-dent: the system experiences a coupling to the environmentthat fluctuates. The correlation time τB of the time-dependentspin-bath operators I

jz (t) = e−iHEt I

jz eiHEt is defined by the

decay to 1/e of the correlation function

ijz (t) = Tr{I

jz (0)I j

z (t)}

Tr{I

jz (0)I j

z (0)} . (13)

Considering that all the bath spins are equivalent, the lattercorrelation functions are identical for every j and share thesame correlation time τB .

III. DYNAMICAL DECOUPLING

A. Notation

The aim of dynamical decoupling is the reduction of theinteraction of the qubit system with the environment, thusretaining the quantum information for as long as possible. Inthe context of DD, it is assumed that it is possible to applyarbitrary single-qubit operations to the system qubit, but that itis not possible to control the environment. One thus applies tothe system short, strong pulses, whose effect can be describedas a refocusing of the system-environment interaction by thecontrol Hamiltonians HC(S)(t) [8,9].

Let us refer to Fig. 1 and consider a single cycle of thesequence having a period τc. In the rotating frame, the operatorthat describes the evolution of the total system from 0 to τc is

U (τc) = Uf (τN+1)N∏

i=1

U iC(τp)Uf (τi), (14)

where from Eq. (8) the free evolution operator is

Uf (t) = exp{−iHf t} (15)

τc

τp

τ1 τi τN+1

… …

τp τp τp

t1 ti-1 ti tN t0

M

tN+1

FIG. 1. Schematic representation of dynamical decoupling. Thesolid boxes represents the control pulses.

and the control evolution operators that act during the time τp

is

U iC(τp) = T exp

{−i

∫ τp

0dt ′

(Hf + Hi

C(S)(t′))}

(16)

with T the Dyson time-ordering operator [42,43]. We assumethat the free evolution Hamiltonian is constant, while the con-trol Hamiltonian Hi

C(S)(t) is constant during τp but changes fordifferent i. The delay times between the control Hamiltoniansare τi = ti − (ti−1 + τp) for i = 2,..,N + 1 and τ1 = t1 − t0,where t0 = 0, tN+1 = τc, and ti represents the time at whichthe i th control operation starts. Figure 1 shows a graphicalrepresentation of these definitions.

Like any unitary evolution, the total propagator can bewritten as the exponential of a Hermitian operator,

U (t) = e−iHeff t . (17)

Using average Hamiltonian theory [16] we can calculate theeffective Hamiltonian Heff as a series expansion,

Heff = H(0) + H(1) + H(2) + · · · =∞∑

n=0

H(n). (18)

The zero-order term H(0) is given by the time integral of thetotal Hamiltonian from time 0 to τc. An ideal DD sequencemakes H(0) = HE , i.e., for ideal pulses, the interactionHamiltonian vanishes to zeroth order. In the Magnus expansion[44], higher-order terms are proportional to increasing powersof τc/τB , since we assume that the environment is weaklycoupled to the system (bSj τB � 1) and in consequence τB isthe dominant time scale [34].

If the basic cycle is iterated M times (see Fig. 1), the totalevolution operator becomes

U (t = Mτc) = [U (τc)]M. (19)

B. Ideal and real pulses

The usual approximation of hard pulses having a radio-frequency field ωp � bSj and duration τp � d−1

i,j , b−1Sj implies

that we can neglect the free precession Hamiltonian andEq. (16) simplifies to

U iC(τp) = exp{−iSuθp} (20)

in the rotating frame, where u = x,y,z and θp = ωpτp is therotation angle around the u axis. In what follows, we shalldenote perfect instantaneous π pulses along x and y by X =exp{−iSxπ} and Y = exp{−iSyπ}, respectively, and a freeevolution of duration τ by fτ .

To take the effect of nonideal pulses into account, one needsto consider errors in the axis and angle of rotation. We write theresulting control propagator as the product of the ideal pulserotation times an error rotation exp{−iSei

θi,e}:U i

C(τp) = exp{−iSeiθi,e} exp{−iSui

θp}. (21)

The total evolution operator is thus

U (τC) = U ′fN+1

(τN+1,τp)N∏

i=1

U iC(0)U ′

fi(τi,τp), (22)

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PERFORMANCE COMPARISON OF DYNAMICAL . . . PHYSICAL REVIEW A 82, 042306 (2010)

where the evolution operators

U ′fi

(τi,τp) = Uf (τi) exp{−iSeiθi,e} (23)

represent a modified free evolution. Note that U ′f1

(τi,τp) =Uf (τ1).

The zero-order average Hamiltonian of the free evolutionperiods (23) for nonperfect pulses is equivalent to interactionsof the general form

Hnpp

SE = axSx + aySy + azSz +∑

j

(bx

Sj Sx + by

Sj Sy + bzSj Sz

)I jz

=∑

u=x,y,z

Su

∑j

(au + bu

Sj Ijz

), (24)

where au and buSj give the renormalized offsets and couplings

respectively, which include the errors of the control Hamilto-nians. This picture can also consider errors of control pulseswhen τp is comparable with the inverse couplings of the freeevolution Hamiltonian.

We now discuss some DD schemes that refocus the system-environment interaction. In all these cases, we assume that thesystem is initially prepared in a coherent superposition of thecomputational basis states. We will refer to the initial state ofthe qubit as Sx , Sy , or Sz, as shown in Fig. 2(a).

C. Hahn echo

The Hahn spin-echo experiment [10] is the pioneer dynam-ical decoupling method and the building block for newer DDproposals. It consists of the application of a π pulse to the S

spin along an axis (say y) transverse to the static field B0 at time

φ1 φ2

y x x y

τ τ

τ τ /2

τ

τ /2

y

Initial state preparation

M

M

(a)

x y y x Cn-1

M

Cn-1 Cn-1 Cn-1

M y y y y τ1 τ2 τ3 τ4 τ5

-x

Sy Sz

y

Sx Sz

Sz

DD sequence

(c)

(b)

(d)

(e)

(f)

FIG. 2. Schemes of dynamical decoupling pulse sequences.Empty and solid rectangles represent π/2 and π pulses, respectively.M represents the number of iterations of the cycle. (a) Initial statepreparation before application of the DD sequence. (b) Hahn spin-echo sequence. (c) CPMG (φ2 = φ1) and CPMG-2 (φ2 = φ1 + π )sequences. (d) PDD sequence. (e) CDD sequence of order n, CDDn =Cn. (f) UDD sequence scheme with four pulses, i.e., UDD of order 4,UDD4.

τ causing an echo at time 2τ [Fig. 2(b)]. The total evolutionoperator can be summarized as fτ Y fτ where the total time(assuming a delta-function pulse) is 2τ . As a consequence, thezero-order average Hamiltonian is

H(0)Hahn = 1

2τ

∫ 2τ

0dt ′H(t ′) = (τHSE − τHSE)

2τ= 0. (25)

The resulting system evolution operator approaches theidentity to within O((τc/τB)2). Thus if τc � τB , a perfectecho (time reversion) is achieved at the total evolution timet = 2τ = τc. When τc is comparable to or longer than τB , theecho decays due to the higher order terms.

D. Carr-Purcell and Carr-Purcell-Meiboom-Gill

To avoid the decay of the echo due to the finite correlationtime of the environment, Carr and Purcell [11] reduced thecycle time by splitting the total time into shorter segmentsof equal length and a refocusing pulse in the middle of eachsegment.

Figure 2(c) shows the pulse sequence for an initial conditionof Sx with φ1 = φ2 = y. The resulting evolution operator isfτ/2Y fτ Y fτ/2. Later on, Meiboom and Gill [12] suggested toshift the phase of the refocusing pulses by π/2, so the rotationaxis is the same as the orientation of the initial state. For perfectpulses, both cases are equivalent, but only the CPMG versioncompensates flip-angle errors of the refocusing pulses. For aflip-angle error exp{−iSei

θi,e} = exp{−iSy�ω1τp} for every i

in Eq. (23), the zero-order average Hamiltonian is proportionalto �ω1Sy . It thus commutes with an initial condition along they axis, (the CPMG case) and has no effect, but it causes anunwanted rotation of an initial state ∝ Sx (the CP case). Inthe following we call this sequence with identical π pulsesCPMG.

An alternative sequence that also compensates flip-angleerrors of the refocusing pulses is shown in Fig. 2(c), withφ2 = φ1 + π = −y. For hard pulses and vanishing delaysbetween the pulses, the zero-order average Hamiltonian ofthis sequence vanishes, for arbitrary flip-angle errors, and thefirst nonvanishing term is of order τc/τB and proportional toSx . As a consequence, an initial condition proportional to Sx

is less affected under this sequence. In what follows, we willcall this DD sequence CPMG-2.

The effect of pulse errors during CPMG and CPMG-2 onthe spin dynamics was studied in Refs. [45–51], and we willshow some effects in the following sections.

E. Periodic dynamical decoupling

A sequence called XY-4 in the NMR community wasproposed initially to compensate the sensitivity of the CPMG-like sequences to nonperfect pulses [13,14]. Later, it wasfound equivalent to the shortest universal DD sequence thatcancels the zero-order average Hamiltonian for a general SEinteraction of the form (24) [34,52]. This sequence, depicted inFig. 2(d) and called periodic dynamical decoupling (PDD), hasan evolution operator of the form Y fτ Xfτ Y fτ Xfτ . Becauseit suppresses SE interactions of the form (24), it compensateserrors of nonideal pulses at the end of the cycle.

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ALVAREZ, AJOY, PENG, AND SUTER PHYSICAL REVIEW A 82, 042306 (2010)

F. Concatenated dynamical decoupling

The CDD scheme [17,34] recursively concatenates lower-order sequences to effectively increase the decoupling order.The CDD evolution operator for a recursion order of n is givenby

CDDn = Cn = YCn−1XCn−1YCn−1XCn−1, (26)

where C0 = fτ and CDD1 = PDD. Figure 2(e) shows ageneral scheme for this process. Each level of concatenationreduces the norm of the first nonvanishing order term of theMagnus expansion of the previous level, provided that thenorm was small enough to begin with. The latter reduction isat the expense of an extension of the cycle time by a factor offour.

G. Uhrig dynamical decoupling (UDD)

Uhrig proposed a different approach to the goal of keepinga qubit alive [20,53]: For a given number N of pulses duringa total time τc, at what times should these pulses be appliedto minimize the effect of the system-environment interaction?The solution he found for the times ti is

ti = τc sin2

[πi

2(N + 1)

], (27)

where tN+1 = τc is the cycle time and t0 = 0 the startingtime. Defining τi = ti − ti−1 the UDD evolution operator fora sequence of N pulses is

UDDN = fτN+1 Y fτNY · · · Y fτ2 Y fτ1 (28)

and its schematic representation is given in Fig. 2(f). TheCPMG sequence is the simplest UDD sequence of orderN = 2.

Cywinski et al. explained the performance of the DDsequence by finding its spectral filter for the bath modes [54].They found that the effect of the UDD pulse sequence leadsto an efficient spectral filter for slow motion bath modes. Itwas shown that UDD is the best sequence for reducing theSE interaction in the limit of low-frequency noise [20,53,54].Rigorous performance bounds for the UDD sequence werefound by Uhrig and Lidar in Ref. [55].

IV. EXPERIMENTAL RESULTS

A. System and environment

Experiments were performed on a polycrystalline adaman-tane sample using a home-built solid-state NMR spectrometerwith a 1H resonance frequency of 300 MHz. The adamantanemolecule contains two nonequivalent carbon atoms. Under ourconditions, they have similar dynamics. Working with naturalabundance (1.1%), the interaction between the 13C-nuclearspins can be neglected. The main mechanism for decoherenceis the interaction with the proton spins. As discussed inSec. II, this interaction (12) is not static, since the dipole-dipolecouplings within the proton bath cause flip-flops of the protonscoupled to the carbon.

Considering that all the proton spins I are equivalent, wecan estimate the correlation time of the time-dependent SEinteraction (12) with the decay time of the correlation function

FIG. 3. (Color online) Evolution of the normalized spin correla-tion functions for the bath spins (protons). The solid line representsthe proton FID signal [ix(t)] and the dashed line the numericallysimulated iz(t).

ijz (t). While the correlation functions i

jz (t) of Eq. (13) cannot

be measured directly because we cannot address individualspins of the bath, we get a very good estimate by measuring

ix(t) = Tr{Ix(0)Ix(t)}Tr{Ix(0)Ix(0)} , (29)

i.e., the proton free-induction decay (FID) (solid line in Fig. 3).The time evolutions in equation (29) are determined by thebath Hamiltonian HE , Ix(t) = e−iHEt Ixe

iHEt and Ix = ∑j I

jx .

Simulating it with the dipole-dipole Hamiltonian of Eq. (11)and using the same Hamiltonian for calculating i

jz (t), we find

the correlation function represented with dashed line in Fig. 3.The spectral density of the bath is well approximated by anormal (Gaussian) distribution and the system-environmentinteraction is weak compared with the intrabath interaction(|bSj |τB <∼ 1/3).

The π pulses for DD were applied on resonance with the13C spins. Their radiofrequency (rf) field of 2π × 48 kHzgives a π-pulse length of τp = 10.4 µs. The measured rffield inhomogeneity is about 10%. We performed experimentswhere the delay τ between successive DD pulses was variedfrom 10 to 200 µs. We prepared the initial state by usingthe sequences of Fig. 2(a) and we measured the survivalprobability of the magnetization

su(t) = Tr{Su(0)Su(t)}Tr{Su(0)Su(0)} , (30)

where u = x,y,z. The solid line of Fig. 4 shows the experi-mental observation of this survival probability from an initialcondition Sx under a free evolution (13C FID).

B. Hahn echo

As shown in Fig. 4, the decay of the S-spin magnetizationis reduced by the Hahn-echo sequence. The results of theHahn echo are marked by square points. Compared to the freeinduction decay, the decay rate is reduced approximately by afactor of 2.

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FIG. 4. (Color online) Survival probability of the S-spin underfree evolution (13C FID) and after a Hahn-echo sequence. An initialcondition Sx was prepared.

C. CPMG

Figure 5 shows the experimental results of the CPMGsequence of Fig. 2(c) with φ1 = φ2 = y. Different rowscorrespond to different initial conditions Sx , Sy , and Sz

of the 13C qubit. The left-hand panels show the survivalprobability (30) as a function of the total evolution timet = Mτc (including the pulses), and the right-hand panels showthe same data as a function of the number of applied pulses.

()

()

()

FIG. 5. (Color online) Magnetization evolution for the CPMGsequence. From top to bottom the initial conditions are Sy , Sx , and Sz.The left-hand panels represent the 13C magnetization as a function ofthe total evolution time while right-hand panels show its evolution asa function of the number of applied pulses. The legend at the bottomgives the delays τ between successive pulses.

The plots show that the decay of the survival probabilitydepends crucially on the initial state of the qubit; we shallhenceforth refer to the initial state in the direction of the DDpulses as the “longitudinal” state and the ones perpendicularto the pulses as the “transverse” states. Flip-angle errors,which arise from inhomogeneous radiofrequency fields, areusually the dominant imperfection in this type of experiments.When the CPMG sequence is applied to a longitudinal initialcondition, flip-angle errors do not affect the performance ofthe decoupling, since they are compensated over each cycleconsisting of two pulses [12]. As a result, the decay ratesfor longitudinal states are about an order of magnitude lowerthan for transverse initial conditions. We also observe anunexpected oscillation pattern for transverse initial states.These kind of strong asymmetries have been reported indifferent samples and have been hypothesized to be due tostimulated echoes induced by pulse errors [45,47,50,51] ordue to the non-negligible effects of the interaction Hamiltonianacting during the finite width pulses [46,48,49].

The right-hand panels show the same data but plottedagainst the number of pulses. They clearly show that theoscillation frequency depends on the number of appliedpulses or equivalently on the total pulse-irradiation time.Similar oscillations have been also reported in differentsamples [46,48–50]. In our experiments, the oscillation patternoriginates from the bimodal distribution of rf field amplitudesin the coil. For our present analysis, the beating is not importantbecause it could be reversed [49] or avoided by improving therf field coil. We instead concentrate on the decay of the enve-lope, which represents the overall survival probability of thesignal.

For the longitudinal initial state (upper panel), Fig. 5(b)shows that the signal decay, as a function of the distance τ

between successive pulses, remains constant until τ = 30 µs.The corresponding cycle time is τc = 2τ + 2τp = 80.8 µs,which is comparable to the bath correlation time τB ; hence;the signal decay for cases below τ = 30 µs is mainly due topulse errors. For longer delays τc > τB (τB ∼ 110 µs), thedecay rate increases because of the reduction of time reversalefficiency in the fluctuating environment.

D. CPMG-2

Figure 6 shows the corresponding results for the CPMG-2sequence. Since the first nonvanishing order of the Magnusexpansion for the CPMG-2 sequence (considering flip-angleerrors) commutes with Sx , we expect that the signal decay forthe Sx initial state is similar to that of the longitudinal initialstate of the CPMG experiments. The experimental resultsshown in Fig. 6 clearly agree with this expectation. For theother initial states, an oscillatory behavior similar to that forthe transverse state of the CPMG is observed. Although theoscillation still depends on the number of pulses (right panels),the frequency is slower than in the CPMG case, and theenvelope of the oscillations decays more slowly. The originof the oscillation pattern is again the bimodal distributionof the inhomogeneity of the rf field generating an effectivefield along the x axis. The experimentally observed oscillationagrees with the results of the effective nutation experiment.Again, we will concentrate on the decay of the envelope.

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()

()

()

FIG. 6. (Color online) Magnetization evolution for the CPMG-2sequence. From top to bottom the initial conditions are Sx , Sy , and Sz.The left-hand panels represent the 13C magnetization as a function ofthe total evolution time, while the right-hand panels show its evolutionas a function of the number of applied pulses. The legend at the bottomgives the delays τ between successive pulses.

E. PDD

Figure 7 shows the signal decay for different initial condi-tions under the application of the PDD sequence of Fig. 2(d).One observes that the signal decay evolves qualitatively similarfor initial conditions in the plane transverse to the static field,i.e., Sx and Sy . This agrees with the theoretical predictions: Thesequence of evolutions Y fτ Xfτ Y fτ Xfτ is nearly symmetricwith respect to x vs. y. The decays still contain a smalloscillatory contribution. Since it appears to depend mostlyon the number of pulses, rather than on the delays betweenthem, we attribute them to pulse errors that are not completelycanceled. Compared to CPMG and CPMG-2, the period ofthe oscillation is one order of magnitude longer, indicatingthat the effect of the pulse imperfections has been reducedby an order of magnitude. This general improvement againstpulse errors is because the sequence cancels the zeroth-orderaverage Hamiltonian of the more general SE interaction (24),while the CPMG sequences cancel only its pure-dephasingpart.

In Figs. 7(b) and 7(d), the decay rates, in units of pulses,up to τ = 40 µs, i.e., τc = 4(τ + τp) = 201.6 µs, are equalto within experimental error. This shows that the sequence ismore robust, compared to previous sequences, in the regimewhere τc exceeds the bath-correlation time. However, from thetime evolution of Figs. 7(a) and 7(c), the decay rate is largerthan the longitudinal case of CPMG or the Sx case of CPMG-2.

()

()

()

FIG. 7. (Color online) Signal decay of the initial state of the qubitfor the PDD sequence. From top to bottom the initial conditions areSx , Sy , and Sz. The left-hand panels represent the decay as a functionof the total evolution time while the right-hand panels show the decayas a function of the number of applied pulses. The legend at the bottomgives the delays τ between successive pulses.

If the initial state is proportional to Sz, its evolution is quali-tatively different. We believe that this results from the fact thatit is parallel to the static field and commutes with the free pre-cession Hamiltonian (8). As a consequence, this evolution re-flects the implementation errors of the sequence. The source ofthe decay is the pulse errors due to which the average Hamilto-nian no longer commutes with the initial state. This experimentprovides a means of quantifying pulse errors, and calibratingan optimal setup of the sequence to enhance its performance.

F. CDD

The qualitative behavior of the CDD experiments is similarbetween different orders and to the PDD one, but they changein the time scale for which the initial state can be maintained.They are also more robust against pulse errors—the oscillationpattern is not observed. For details of the survival probabilityevolution see Appendix. A summary of the results is presentedin Fig. 8 in Sec. V where the decay times for different CDDorders and delays τ between pulses are plotted.

G. UDD

The experimental survival probabilities for the UDD se-quences have the same qualitative behavior as the CPMGcurves. They manifest the same asymmetries with respect tothe initial state. That is expected because UDD sequences also

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FIG. 8. (Color online) Relaxation times of different initial condi-tions under DD conditions as a function of the delay between pulsesτ (left panels) and the cycle time τc (right panels). From top to bottomthe initial condition is given by ρ0 = Sx , Sy , and Sz respectively. Anoptimal τ and consequently τc is observed for each sequence. Thereduction of the relaxation time to the right side of the optimal valueis due to the shifting environment: in this regime the cycle time islonger than the correlation time of the bath, τc > τB . The reductionfor short cycle times indicates that in this regime, accumulated pulseerrors dominate.

only reduce SE interactions of the form (10). We observedthat with increasing UDD order, the decay rates increase,as predicted in Ref. [55] for the conditions satisfied in ourexperiments where |bSj |τB < 1 and τc ∼ τB . A summaryof the rates is shown in the next section in Fig. 8 wherewe have used τ as the average delay between pulses. Anextensive analysis of the performance of UDD sequences andnonequidistant pulse sequences against equidistant ones for thepresent experimental conditions will be given elsewhere [56].

V. COMPARISONS: OPTIMAL CHOICES

The goal of DD is the preservation of quantum states bythe application of suitable decoupling sequences. If the pulsesare ideal and they are applied with very short delays, it ispossible to preserve quantum states for arbitrarily long times inthe presence of a system-environment coupling that is linear inthe system operators. However, for experiments using nonidealpulses, a finite cycle time optimizes the DD performance. This

can be seen very clearly in the summary of the experimentalresults presented in Fig. 8. The left panels show the DD decaytimes as a function of the delay τ for every sequence anddifferent initial conditions. As an example, for CPMG when theinitial condition is longitudinal to the pulses (Sy), the optimalcycle time is τc = 80.8 µs (τ = 30 µs). For longer cycle times,the decay time gets shorter, since the environment changesduring the cycle and the refocusing efficiency decreases.

While shorter cycle times should give even better resultsunder ideal conditions, we find experimentally a decrease ofthe relaxation time. This can be attributed to an accumulationof pulse errors, which dominates in this regime. Similar resultsare observed for the CPMG-2 if we exchange Sy with Sx . Thiscan be seen clearly under conditions of transverse initial states(Sx for CPMG) where the decay time is proportional to thecycle time. This means that the error per cycle is independentof the cycle time and corresponds thus to a zero-order termof the average Hamiltonian. This is the behavior expected forflip angle errors, which are the main source of the decay inthis regime. Since flip angle errors are in no way compensatedfor transverse states in the CPMG sequence, their accumulatedeffect is so strong that the optimal cycle time exceeds τB and thesequence performs only marginally better than the Hahn-echosequence, which has the longest cycle time.

If we consider the CPMG-2 sequence with the initialconditions Sy and Sz, the decay time grows ∝ τ 2

c for shorttimes. This implies that in this case, the dominant error term isproportional to τc, i.e., it corresponds to a first-order term of theaverage Hamiltonian. Moreover, its optimal relaxation time isone order of magnitude longer than the Hahn-echo decay time.

The behavior of the UDD sequences is similar to that ofCPMG. We show here only their decay times for Sy as initialcondition [Fig. 8(c)], i.e., longitudinal to the DD pulses. Theyare plotted as a function of the average delay between pulses.The figure shows that the UDD decay times are always shorterthan those of CPMG. Moreover, increasing the UDD orderreduces the decay time, as expected by theoretical expectationswhen |bSj |τB < 1 and τc ∼ τB [55]. A regime where UDDperforms better than CPMG may perhaps exist at short cycletimes compared with τB [56], provided the pulse errors can bemade sufficiently small that they do not dominate over externalsources of decoherence. Recent proposals of UDD basedsequences that reduce decoherence of a general SE interaction,like Eq. (24), could allow one to find this regime [21,22].

For PDD the optimal cycle time for Sx and Sy is τc =321.6 µs, (τ = 70 µs), which is longer than τB . The resultingperformance is relatively poor. The resulting decay time issimilar to that of CPMG for the same cycle time [Fig. 8(d)] orCPMG-2 for the Sx initial condition [Fig. 8(b)]. However, forthese particular initial conditions CPMG and CPMG-2 can bemade to perform an order of magnitude better by reducing thecycle time.

For CDD2 with initial conditions transverse to the staticfield, the optimal cycle time is τc = 16τ + 20τp = 688 µs,i.e., τ = 30 µs. For CDD3 and CDD4 the shortest delay timebetween pulses of τ = 10 µs and τ = 2.5 µs are the optimalsituations, giving τc(CDD3) = 64τ + 84τp = 1513.6 µs andτc(CDD4) = 256τ + 388τp = 4675.2 µs. The optimal delay τ

becomes shorter with increasing CDD order, because the cycletime increases by a factor of 4 for each level of concatenation.

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FIG. 9. (Color online) CDD order n as a function of its optimaldelay between pulses τopt to reduce decoherence. The experimentalsquare points seem to satisfy a relation given by n = c − b ln(τopt/τB ).The solid (red) line shows a fitting curve with parameters c = (0.9 ±0.2) and b = (−0.9 ± 0.1).

Apparently, the pulse errors do not accumulate as strongly asin the case of CPMG, which may be attributed to the factthat CDD is designed to compensate pulse errors [17,34].The crossover cycle time, where the transition occurs froma decay dominated by pulse errors to the regime wherethe decay is dominated by the short bath correlation timeis increased, as shown in the right-hand panels. Even forcycle times that are much longer than the bath-correlationtime, the CDD provides a significant reduction of the deco-herence rate compared to the free evolution decay (dottedlines) and the Hahn-echo decay (dashed lines). An optimalcycle time when considering imperfect finite pulses waspredicted by Khodjasteh and Lidar [34]. Figure 9 shows theexperimental relation between the optimal delays τ and theirrespective CDD order n (square points). It seems to satisfy arelation given by n = c − b ln(τopt/τB ), where c and b areconstants and τopt is the optimal delay for a given n (seebelow).

The unifying result of the curves shown in the left-handpanels of Fig. 8 is that the optimal delay between pulses isalways shorter than the bath correlation time, with comparablevalues for all sequences, with the single exception of theCPMG sequence for initial conditions Sx and Sz, as discussedabove. Clearly, this time scale is determined by the (average)delay between pulses τ , not by the cycle time τc. Expressingthis differently, one might say that only a small fraction ofwhat is lost in a single echo can be refocused by compensatedsequences. If we look at pulse spacings longer than the bathcorrelation time τB , the differences between sequences becomevery small and the decay times approximate those of FIDand Hahn echo. Accordingly, it appears important to keepthe number of pulses per unit time constant when comparingdifferent DD sequences.

We now compare the different DD sequences with theoptimal delay between pulses for each sequence. Figure 10shows the evolution of the survival probabilities for differentinitial conditions for all the sequences discussed here.

As a general rule, we note that for increasing CDD order,the optimal pulse delay τ gets shorter and for longer delays

FIG. 10. (Color online) Time evolution of the survival probabilityfor the optimal delay between pulses of the different DD sequences.From top to bottom the initial states are Sx , Sy , and Sz. The optimaldelays τ are given in the legends.

between pulses, higher CDD orders do not perform betterthan lower CDD orders. Hence, keeping the delay betweenpulses constant, there is an optimal CDD order for reducingdecoherence as predicted in Refs. [34,35]. It is difficult tofind accurately the optimal CDD order as a function ofτ from Figs. 8(a) and 8(c) in order to compare with thetheoretical predictions of Eq. (140) in Ref. [35]. However,the relation given in Fig. 9 for the optimal delay betweenpulses τopt behaves similar. A linear fitting of the experimentaldata gives n = c − b ln(τopt/τB) with c = (0.9 ± 0.2) and b =(−0.9 ± 0.1) agreeing well with the predicted expression [35].

The best DD sequence and its corresponding optimal cycletime depends on hardware limitations and importantly on thedesired goal. If one aims to freeze a quantum state duringa short time, its value will bound the cycle time and as aresult, the maximal CDD order that can be applied. For longertimes, increasing the CDD order will be advantageous, butpower dissipation may force a reduction in the number ofpulses and simultaneously the CDD order. For specific initialconditions, CPMG and CPMG-2 are the best choices forreducing decoherence; however, the large asymmetry of thesesequences to other initial conditions limits their usefulnesswhen the initial state of the qubit is unknown. Note that inthese cases, it has been shown that coherences could be frozenas labeled polarization [51]. If the goal is the preservation ofan unknown quantum state, the CDD sequences provide thebest overall performance.

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While the SE interaction produces pure dephasing tothe spin, in principle DD sequences that compensate puredephasing decoherence should be sufficient. In conse-quence concatenating sequences like fτ Y fτ Y , which reducepure dephasing processes, could be beneficial. However,the finite precision of control pulses generates an effectiveHamiltonian of the form (24), and thus sequences developedto reduce pure dephasing processes have asymmetric perfor-mances against initial states directions, i.e., they do not gener-ate a unit evolution operator of the qubit. Thus, we compared,as a test bed, CPMG/UDD sequence that compensate puredephasing with XY-4 [13,14] based sequences that compensatea general interaction in order to show their effects against pulseerrors.

From our results it is evident that for short delays themain source of DD decays are static pulse errors; this limitsthe maximal performance. However, CPMG and CPMG-2show the potentially achievable DD performance if the pulseerrors are reduced. This implies that new DD proposals shouldfocus on the compensation of pulse errors for these kinds ofexperimental conditions, similar to the proposal by Viola andKnill [15] or Uhrig and Pasini [57,58]. CDD-type sequencesdo compensate for pulse errors but only at the end of the CDDcycle. This limits their performance because of the exponentialgrowth of the cycle time with the CDD order. As an alternativemethod, we suggest to find sequences that compensate pulseerrors to zero order during each step of the concatenationprocedure. That would be advantageous because the zero-ordercompensation cycle time remains constant and equal to thePDD cycle time, as assumed for ideal pulses.

VI. CONCLUSIONS

We have experimentally applied different dynamical de-coupling sequences to a qubit-system coupled to a spin-bathin order to test and compare their performance. The systemused is typical for spin-based solid-state systems where thespectral density of the bath is given by a normal (Gaussian)distribution and the system-environment interaction is weakcompared with the intrabath interaction. The experiments wereperformed in the regime where the average spacing betweenthe pulses is comparable to the bath-correlation time. Thisarticle focuses on measuring and fighting decoherence, andthe results do not depend on the readout or initializationscheme used for that purpose. Thus, the results should applydirectly to other spin-based quantum information process-ing systems, such as electron spins in diamonds [30–32],electron spins in quantum dots [28,29,39], and donors insilicon [40,41].

While the design of DD sequences is typically basedon the assumption that the cycle times is shorter than thebath-correlation time, we demonstrated that even withoutsatisfying this condition, dynamical decoupling reduces de-coherence significantly. We showed that the main limitation tothe reduction of the DD decay rates is due to the finite precisionof the control operations: In our system, flip-angle errors werethe main source. Therefore, CPMG or UDD-type sequencesthat reduce purely dephasing or purely dissipative interactionswith the bath perform well only for specific initial conditionsof the qubit ensemble. For the privileged initial condition,

CPMG-type sequences performed better than any other DDsequence. But, if the goal is to approach a unit evolutionoperator, PDD sequence and its concatenated form (CDD) arethe best overall option. In agreement with previous predictions[34–36] our results show that, depending on limitations ofhardware and the required time to keep the initial statecoherent, increasing the CDD order is not always useful. Thereis an optimal CDD order depending on the power availablefor the control pulses and their finite precision. We presentstrong evidence that in order to improve dynamical decouplingsequences, they should be designed to compensate pulseerrors.

ACKNOWLEDGMENTS

This work is supported by the DFG through Su 192/24-1.G.A.A. thanks the Alexander von Humboldt Foundation. Wethank Daniel Lidar for helpful discussions and Marko Lovricand Ingo Niemeyer for technical support.

APPENDIX: CDD EXPERIMENTS

Figures 11, 12, and 13 show the experimentally observedsignal decays for CDD2, CDD3, and CDD4 respectively.

()

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FIG. 11. (Color online) Signal decay of the initial state of thequbit for the CDD2 sequence. From top to bottom the initial conditionsare Sx , Sy , and Sz. The left-hand panels represent the decay asa function of the total evolution time while the right-hand panelsshow the decay as a function of the number of applied pulses.The legend at the bottom gives the delays τ between successivepulses.

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FIG. 12. (Color online) Signal decay of the initial state of thequbit for the CDD3 sequence. From top to bottom the initial conditionsare Sx , Sy , and Sz. The left-hand panels represent the decay as afunction of the total evolution time while right-hand panels the decayas a function of the number of applied pulses. The legend at thebottom gives the delays τ between successive pulses.

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FIG. 13. (Color online) Signal decay of the initial state of thequbit for the CDD4 sequence. From top to bottom the initial conditionsare Sx , Sy , and Sz. The left-hand panels represent the decay as afunction of the total evolution time while the right-hand panels showthe decay as a function of the number of applied pulses. The legendat the bottom gives the delays τ between successive pulses.

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